Subgroups — C₃²×#3274 #11770323

Order 1152.

1668 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
2	C₃²×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×D₄²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#469	C₂	1	2	yes	maximal, complemented
6	C₃²×#460	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#459	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#449	C₂	1	2	yes	maximal, complemented
9	C₃²×#447	C₂	1	2	yes	maximal, complemented
10	C₃²×C₄×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃²×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃×#3274	C₃	1	4	yes	maximal, direct product factor
13	C₂×C₃²×#41	C₂²	1	2	yes	complemented
14	C₂²×C₃²×D₄	—	2	1	no	—
15	C₂²×C₃²×D₄	C₂²	1	2	yes	—
16	C₃²×AGL(1,ℤ/8)	—	2	8	no	—
17	C₂×C₃²×SD₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₃²×SD₁₆	C₂²	1	2	yes	complemented
19	C₂×C₃²×D₈	C₂²	1	2	yes	—
20	C₂×C₃²×D₈	C₂²	1	2	yes	complemented
21	C₂×C₃²×D₈	C₂²	1	2	yes	complemented
22	C₂×C₃²×M₁₆	C₂²	1	2	yes	complemented
23	C₃²×#127	—	2	1	no	—
24	C₃²×#127	C₂²	1	1	yes	characteristic
25	C₃²×#127	C₂²	1	1	yes	characteristic, complemented
26	C₃²×#121	—	2	1	no	—
27	C₃²×#121	—	2	2	no	—
28	C₃²×#121	C₂²	1	2	yes	—
29	C₃²×#121	C₂²	1	2	yes	complemented
30	C₃²×#120	—	2	2	no	—
31	C₃²×C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
32	C₃²×C₄×D₄	—	2	1	no	—
33	C₃²×C₄×D₄	C₂²	1	1	yes	characteristic
34	C₃²×C₄×D₄	C₂²	1	1	yes	characteristic
35	C₃²×#106	C₂²	1	1	yes	characteristic, complemented
36	C₃²×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
37	C₃²×#103	C₂²	1	1	yes	characteristic, complemented
38	C₃²×#102	C₂²	1	1	yes	characteristic
39	C₃²×#102	C₂²	1	2	yes	complemented
40	C₃²×#102	C₂²	1	2	yes	complemented
41	C₃²×C₂²⋊C₈	C₂²	1	2	yes	complemented
42	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic, complemented
43	C₂×C₃×AGL(1,ℤ/8)	C₂×C₃	1	8	yes	complemented
44	C₃×#550	C₂×C₃	1	4	yes	complemented
45	C₃×D₄²	C₂×C₃	1	4	yes	complemented
46	C₃×#493	C₂×C₃	1	4	yes	complemented
47	C₃×#469	C₂×C₃	1	8	yes	complemented
48	C₃×#460	C₂×C₃	1	4	yes	complemented
49	C₃×#459	C₂×C₃	1	4	yes	complemented
50	C₃×#449	C₂×C₃	1	8	yes	complemented
51	C₃×#447	C₂×C₃	1	8	yes	complemented
52	C₃×C₄×SD₁₆	C₂×C₃	1	4	yes	complemented
53	C₃×#436	C₂×C₃	1	4	yes	complemented
54	C₂⁴×C₃²	—	2	2	no	—
55	C₃²×#41	—	2	4	no	—
56	C₃²×#41	—	4	2	no	—
57	C₂×C₃²×Q₈	C₂³	1	1	yes	characteristic
58	C₂×C₃²×D₄	—	2	4	no	—
59	C₂×C₃²×D₄	—	2	2	no	—
60	C₂×C₃²×D₄	—	2	2	no	—
61	C₂×C₃²×D₄	—	2	2	no	—
62	C₂×C₃²×D₄	—	2	1	no	—
63	C₂×C₃²×D₄	—	2	2	no	—
64	C₂×C₃²×D₄	—	2	1	no	—
65	C₂×C₃²×D₄	—	2	1	no	—
66	C₂×C₃²×D₄	—	2	1	no	—
67	C₂×C₃²×D₄	—	4	2	no	—
68	C₂×C₃²×D₄	—	4	4	no	—
69	C₂×C₃²×D₄	D₄	1	1	yes	characteristic
70	C₂×C₃²×D₄	C₂³	1	2	yes	—
71	C₂×C₃²×D₄	C₂³	1	2	yes	—
72	C₂×C₃²×D₄	C₂³	1	1	yes	characteristic
73	C₂²×C₃²×C₄	—	2	1	no	—
74	C₂²×C₃²×C₄	—	2	2	no	—
75	C₂²×C₃²×C₄	C₂³	1	2	yes	—
76	C₃²×SD₁₆	—	2	4	no	—
77	C₃²×SD₁₆	D₄	1	4	yes	complemented
78	C₃²×D₈	—	2	4	no	—
79	C₃²×D₈	—	2	4	no	—
80	C₃²×D₈	—	4	2	no	—
81	C₃²×M₁₆	—	2	4	no	—
82	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
83	C₂×C₃²×C₈	C₂³	1	2	yes	—
84	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
85	C₃²×C₄⋊C₄	—	2	1	no	—
86	C₃²×C₄⋊C₄	D₄	1	1	yes	characteristic
87	C₃²×C₄⋊C₄	C₂³	1	1	yes	characteristic
88	C₃²×C₄⋊C₄	C₂³	1	1	yes	characteristic
89	C₃²×C₂²⋊C₄	—	2	2	no	—
90	C₃²×C₂²⋊C₄	—	2	1	no	—
91	C₃²×C₂²⋊C₄	—	2	2	no	—
92	C₃²×C₂²⋊C₄	D₄	1	2	yes	—
93	C₃²×C₄²	—	2	1	no	—
94	C₃²×C₄²	C₂³	1	1	yes	characteristic
95	#3274	C₃²	1	1	yes	characteristic, direct product factor, Sylow
96	C₂×C₃×#41	C₂²×C₃	1	8	yes	complemented
97	C₂²×C₃×D₄	—	2	4	no	—
98	C₂²×C₃×D₄	C₂²×C₃	1	8	yes	—
99	C₃×AGL(1,ℤ/8)	—	2	32	no	—
100	C₂×C₃×SD₁₆	C₂²×C₃	1	4	yes	—
101	C₂×C₃×SD₁₆	C₂²×C₃	1	8	yes	complemented
102	C₂×C₃×D₈	C₂²×C₃	1	8	yes	—
103	C₂×C₃×D₈	C₂²×C₃	1	8	yes	complemented
104	C₂×C₃×D₈	C₂²×C₃	1	8	yes	complemented
105	C₂×C₃×M₁₆	C₂²×C₃	1	8	yes	complemented
106	C₃×#127	—	2	4	no	—
107	C₃×#127	C₂²×C₃	1	4	yes	—
108	C₃×#127	C₂²×C₃	1	4	yes	complemented
109	C₃×#121	—	2	4	no	—
110	C₃×#121	—	2	8	no	—
111	C₃×#121	C₂²×C₃	1	8	yes	—
112	C₃×#121	C₂²×C₃	1	8	yes	complemented
113	C₃×#120	—	2	8	no	—
114	C₃×C₄×Q₈	C₂²×C₃	1	4	yes	complemented
115	C₃×C₄×D₄	—	2	4	no	—
116	C₃×C₄×D₄	C₂²×C₃	1	4	yes	—
117	C₃×C₄×D₄	C₂²×C₃	1	4	yes	—
118	C₃×#106	C₂²×C₃	1	4	yes	complemented
119	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	complemented
120	C₃×#103	C₂²×C₃	1	4	yes	complemented
121	C₃×#102	C₂²×C₃	1	4	yes	—
122	C₃×#102	C₂²×C₃	1	8	yes	complemented
123	C₃×#102	C₂²×C₃	1	8	yes	complemented
124	C₃×C₂²⋊C₈	C₂²×C₃	1	8	yes	complemented
125	C₃×C₄×C₈	C₂²×C₃	1	4	yes	complemented
126	C₂³×C₃²	—	2	1	no	—
127	C₂³×C₃²	—	2	2	no	—
128	C₂³×C₃²	—	2	2	no	—
129	C₂³×C₃²	—	4	4	no	—
130	C₂³×C₃²	—	4	4	no	—
131	C₂³×C₃²	—	8	2	no	—
132	C₂³×C₃²	C₂×D₄	1	2	yes	—
133	C₃²×Q₈	—	2	1	no	—
134	C₃²×Q₈	C₂×D₄	1	2	yes	complemented
135	C₃²×D₄	—	2	2	no	—
136	C₃²×D₄	—	2	2	no	—
137	C₃²×D₄	—	2	2	no	—
138	C₃²×D₄	—	2	1	no	—
139	C₃²×D₄	—	2	2	no	—
140	C₃²×D₄	—	2	2	no	—
141	C₃²×D₄	—	4	4	no	—
142	C₃²×D₄	—	4	4	no	—
143	C₃²×D₄	—	4	4	no	—
144	C₃²×D₄	—	4	2	no	—
145	C₃²×D₄	—	4	2	no	—
146	C₃²×D₄	—	8	1	no	—
147	C₃²×D₄	—	8	1	no	—
148	C₃²×D₄	—	8	2	no	—
149	C₃²×D₄	C₂×D₄	1	2	yes	—
150	C₂×C₃²×C₄	—	2	1	no	—
151	C₂×C₃²×C₄	—	2	2	no	—
152	C₂×C₃²×C₄	—	2	2	no	—
153	C₂×C₃²×C₄	—	2	1	no	—
154	C₂×C₃²×C₄	—	2	1	no	—
155	C₂×C₃²×C₄	—	4	4	no	—
156	C₂×C₃²×C₄	—	4	2	no	—
157	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
158	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
159	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
160	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
161	C₃²×C₈	—	2	2	no	—
162	C₃²×C₈	—	2	1	no	—
163	C₃²×C₈	C₂×D₄	1	2	yes	complemented
164	C₂×AGL(1,ℤ/8)	C₂×C₃²	1	2	yes	complemented
165	#550	C₂×C₃²	1	1	yes	characteristic, complemented
166	D₄²	C₂×C₃²	1	1	yes	characteristic, complemented
167	#493	C₂×C₃²	1	1	yes	characteristic, complemented
168	#469	C₂×C₃²	1	2	yes	complemented
169	#460	C₂×C₃²	1	1	yes	characteristic, complemented
170	#459	C₂×C₃²	1	1	yes	characteristic, complemented
171	#449	C₂×C₃²	1	2	yes	complemented
172	#447	C₂×C₃²	1	2	yes	complemented
173	C₄×SD₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
174	#436	C₂×C₃²	1	1	yes	characteristic, complemented
175	C₂⁴×C₃	—	2	8	no	—
176	C₃×#41	—	2	16	no	—
177	C₃×#41	—	4	8	no	—
178	C₂×C₃×Q₈	C₂³×C₃	1	4	yes	—
179	C₂×C₃×D₄	—	2	16	no	—
180	C₂×C₃×D₄	—	2	8	no	—
181	C₂×C₃×D₄	—	2	8	no	—
182	C₂×C₃×D₄	—	2	8	no	—
183	C₂×C₃×D₄	—	2	4	no	—
184	C₂×C₃×D₄	—	2	8	no	—
185	C₂×C₃×D₄	—	2	4	no	—
186	C₂×C₃×D₄	—	2	4	no	—
187	C₂×C₃×D₄	—	2	4	no	—
188	C₂×C₃×D₄	—	4	8	no	—
189	C₂×C₃×D₄	—	4	16	no	—
190	C₂×C₃×D₄	C₃×D₄	1	4	yes	—
191	C₂×C₃×D₄	C₂³×C₃	1	8	yes	—
192	C₂×C₃×D₄	C₂³×C₃	1	8	yes	—
193	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
194	C₂²×C₃×C₄	—	2	8	no	—
195	C₂²×C₃×C₄	—	2	4	no	—
196	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
197	C₃×SD₁₆	—	2	16	no	—
198	C₃×SD₁₆	C₃×D₄	1	16	yes	complemented
199	C₃×D₈	—	2	16	no	—
200	C₃×D₈	—	2	16	no	—
201	C₃×D₈	—	4	8	no	—
202	C₃×M₁₆	—	2	16	no	—
203	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
204	C₂×C₃×C₈	C₂³×C₃	1	8	yes	—
205	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
206	C₃×C₄⋊C₄	—	2	4	no	—
207	C₃×C₄⋊C₄	C₃×D₄	1	4	yes	—
208	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
209	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
210	C₃×C₂²⋊C₄	—	2	8	no	—
211	C₃×C₂²⋊C₄	—	2	4	no	—
212	C₃×C₂²⋊C₄	—	2	8	no	—
213	C₃×C₂²⋊C₄	C₃×D₄	1	8	yes	—
214	C₃×C₄²	—	2	4	no	—
215	C₃×C₄²	C₂³×C₃	1	4	yes	—
216	C₂²×C₃²	—	2	2	no	—
217	C₂²×C₃²	—	2	2	no	—
218	C₂²×C₃²	—	2	2	no	—
219	C₂²×C₃²	—	2	2	no	—
220	C₂²×C₃²	—	4	1	no	—
221	C₂²×C₃²	—	4	1	no	—
222	C₂²×C₃²	—	4	2	no	—
223	C₂²×C₃²	—	4	2	no	—
224	C₂²×C₃²	—	4	2	no	—
225	C₂²×C₃²	—	4	2	no	—
226	C₂²×C₃²	—	4	2	no	—
227	C₂²×C₃²	—	4	2	no	—
228	C₂²×C₃²	—	8	8	no	—
229	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
230	C₃²×C₄	—	2	2	no	—
231	C₃²×C₄	—	2	2	no	—
232	C₃²×C₄	—	2	1	no	—
233	C₃²×C₄	—	4	1	no	—
234	C₃²×C₄	—	4	1	no	—
235	C₃²×C₄	AGL(1,ℤ/8)	1	2	yes	complemented
236	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
237	C₃²×C₄	H(5,𝔽₂)	1	1	yes	characteristic
238	C₂×#41	C₂²×C₃²	1	2	yes	complemented
239	C₂²×D₄	—	2	1	no	—
240	C₂²×D₄	C₂²×C₃²	1	2	yes	—
241	AGL(1,ℤ/8)	—	2	8	no	—
242	C₂×SD₁₆	C₂²×C₃²	1	1	yes	characteristic
243	C₂×SD₁₆	C₂²×C₃²	1	2	yes	complemented
244	C₂×D₈	C₂²×C₃²	1	2	yes	—
245	C₂×D₈	C₂²×C₃²	1	2	yes	complemented
246	C₂×D₈	C₂²×C₃²	1	2	yes	complemented
247	C₂×M₁₆	C₂²×C₃²	1	2	yes	complemented
248	#127	—	2	1	no	—
249	#127	C₂²×C₃²	1	1	yes	characteristic
250	#127	C₂²×C₃²	1	1	yes	characteristic, complemented
251	#121	—	2	1	no	—
252	#121	—	2	2	no	—
253	#121	C₂²×C₃²	1	2	yes	—
254	#121	C₂²×C₃²	1	2	yes	complemented
255	#120	—	2	2	no	—
256	C₄×Q₈	C₂²×C₃²	1	1	yes	characteristic, complemented
257	C₄×D₄	—	2	1	no	—
258	C₄×D₄	C₂²×C₃²	1	1	yes	characteristic
259	C₄×D₄	C₂²×C₃²	1	1	yes	characteristic
260	#106	C₂²×C₃²	1	1	yes	characteristic, complemented
261	C₄⋊C₈	C₂²×C₃²	1	1	yes	characteristic, complemented
262	#103	C₂²×C₃²	1	1	yes	characteristic, complemented
263	#102	C₂²×C₃²	1	1	yes	characteristic
264	#102	C₂²×C₃²	1	2	yes	complemented
265	#102	C₂²×C₃²	1	2	yes	complemented
266	C₂²⋊C₈	C₂²×C₃²	1	2	yes	complemented
267	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic, complemented
268	C₂³×C₃	—	2	4	no	—
269	C₂³×C₃	—	2	8	no	—
270	C₂³×C₃	—	2	8	no	—
271	C₂³×C₃	—	4	16	no	—
272	C₂³×C₃	—	4	16	no	—
273	C₂³×C₃	—	8	8	no	—
274	C₂³×C₃	C₂×C₃×D₄	1	8	yes	—
275	C₃×Q₈	—	2	4	no	—
276	C₃×Q₈	C₂×C₃×D₄	1	8	yes	complemented
277	C₃×D₄	—	2	8	no	—
278	C₃×D₄	—	2	8	no	—
279	C₃×D₄	—	2	8	no	—
280	C₃×D₄	—	2	4	no	—
281	C₃×D₄	—	2	8	no	—
282	C₃×D₄	—	2	8	no	—
283	C₃×D₄	—	4	16	no	—
284	C₃×D₄	—	4	16	no	—
285	C₃×D₄	—	4	16	no	—
286	C₃×D₄	—	4	8	no	—
287	C₃×D₄	—	4	8	no	—
288	C₃×D₄	—	8	4	no	—
289	C₃×D₄	—	8	4	no	—
290	C₃×D₄	—	8	8	no	—
291	C₃×D₄	C₂×C₃×D₄	1	8	yes	—
292	C₂×C₃×C₄	—	2	4	no	—
293	C₂×C₃×C₄	—	2	8	no	—
294	C₂×C₃×C₄	—	2	8	no	—
295	C₂×C₃×C₄	—	2	4	no	—
296	C₂×C₃×C₄	—	2	4	no	—
297	C₂×C₃×C₄	—	4	16	no	—
298	C₂×C₃×C₄	—	4	8	no	—
299	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
300	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
301	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
302	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
303	C₃×C₈	—	2	8	no	—
304	C₃×C₈	—	2	4	no	—
305	C₃×C₈	C₂×C₃×D₄	1	8	yes	complemented
306	C₂×C₃²	—	4	2	no	—
307	C₂×C₃²	—	4	2	no	—
308	C₂×C₃²	—	8	2	no	—
309	C₂×C₃²	—	8	2	no	—
310	C₂×C₃²	D₄²	1	1	yes	characteristic
311	C₂×C₃²	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
312	C₂×C₃²	#576	1	1	yes	characteristic
313	C₂⁴	—	2	2	no	—
314	#41	—	2	4	no	—
315	#41	—	4	2	no	—
316	C₂×Q₈	C₂³×C₃²	1	1	yes	characteristic
317	C₂×D₄	—	2	4	no	—
318	C₂×D₄	—	2	2	no	—
319	C₂×D₄	—	2	2	no	—
320	C₂×D₄	—	2	2	no	—
321	C₂×D₄	—	2	1	no	—
322	C₂×D₄	—	2	2	no	—
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	—	2	1	no	—
325	C₂×D₄	—	2	1	no	—
326	C₂×D₄	—	4	2	no	—
327	C₂×D₄	—	4	4	no	—
328	C₂×D₄	C₃²×D₄	1	1	yes	characteristic
329	C₂×D₄	C₂³×C₃²	1	2	yes	—
330	C₂×D₄	C₂³×C₃²	1	2	yes	—
331	C₂×D₄	C₂³×C₃²	1	1	yes	characteristic
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	2	1	no	—
334	C₂²×C₄	C₂³×C₃²	1	2	yes	—
335	SD₁₆	—	2	4	no	—
336	SD₁₆	C₃²×D₄	1	4	yes	complemented
337	D₈	—	2	4	no	—
338	D₈	—	2	4	no	—
339	D₈	—	4	2	no	—
340	M₁₆	—	2	4	no	—
341	C₂×C₈	C₂³×C₃²	1	1	yes	characteristic
342	C₂×C₈	C₂³×C₃²	1	2	yes	—
343	C₂×C₈	C₂³×C₃²	1	1	yes	characteristic
344	C₄⋊C₄	—	2	1	no	—
345	C₄⋊C₄	C₃²×D₄	1	1	yes	characteristic
346	C₄⋊C₄	C₂³×C₃²	1	1	yes	characteristic
347	C₄⋊C₄	C₂³×C₃²	1	1	yes	characteristic
348	C₂²⋊C₄	—	2	2	no	—
349	C₂²⋊C₄	—	2	1	no	—
350	C₂²⋊C₄	—	2	2	no	—
351	C₂²⋊C₄	C₃²×D₄	1	2	yes	—
352	C₄²	—	2	1	no	—
353	C₄²	C₂³×C₃²	1	1	yes	characteristic
354	C₂²×C₃	—	2	8	no	—
355	C₂²×C₃	—	2	8	no	—
356	C₂²×C₃	—	2	8	no	—
357	C₂²×C₃	—	2	8	no	—
358	C₂²×C₃	—	4	4	no	—
359	C₂²×C₃	—	4	4	no	—
360	C₂²×C₃	—	4	8	no	—
361	C₂²×C₃	—	4	8	no	—
362	C₂²×C₃	—	4	8	no	—
363	C₂²×C₃	—	4	8	no	—
364	C₂²×C₃	—	4	8	no	—
365	C₂²×C₃	—	4	8	no	—
366	C₂²×C₃	—	8	32	no	—
367	C₂²×C₃	C₂²×C₃×D₄	1	4	yes	—
368	C₃×C₄	—	2	4	no	—
369	C₃×C₄	—	2	8	no	—
370	C₃×C₄	—	2	8	no	—
371	C₃×C₄	—	4	4	no	—
372	C₃×C₄	—	4	4	no	—
373	C₃×C₄	C₃×AGL(1,ℤ/8)	1	8	yes	complemented
374	C₃×C₄	C₂²×C₃×D₄	1	4	yes	—
375	C₃×C₄	C₃×H(5,𝔽₂)	1	4	yes	—
376	C₃²	#3274	1	1	yes	characteristic, direct product factor, Sylow
377	C₂³	—	2	1	no	—
378	C₂³	—	2	2	no	—
379	C₂³	—	2	2	no	—
380	C₂³	—	4	4	no	—
381	C₂³	—	4	4	no	—
382	C₂³	—	8	2	no	—
383	C₂³	C₂×C₃²×D₄	1	2	yes	—
384	Q₈	—	2	1	no	—
385	Q₈	C₂×C₃²×D₄	1	2	yes	complemented
386	D₄	—	2	2	no	—
387	D₄	—	2	2	no	—
388	D₄	—	2	1	no	—
389	D₄	—	2	2	no	—
390	D₄	—	2	2	no	—
391	D₄	—	2	2	no	—
392	D₄	—	4	4	no	—
393	D₄	—	4	4	no	—
394	D₄	—	4	4	no	—
395	D₄	—	4	2	no	—
396	D₄	—	4	2	no	—
397	D₄	—	8	1	no	—
398	D₄	—	8	1	no	—
399	D₄	—	8	2	no	—
400	D₄	C₂×C₃²×D₄	1	2	yes	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	4	4	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
409	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
410	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
411	C₂×C₄	C₂⁴×C₃²	1	1	yes	characteristic
412	C₈	—	2	2	no	—
413	C₈	—	2	1	no	—
414	C₈	C₂×C₃²×D₄	1	2	yes	complemented
415	C₂×C₃	—	4	8	no	—
416	C₂×C₃	—	4	8	no	—
417	C₂×C₃	—	8	8	no	—
418	C₂×C₃	—	8	8	no	—
419	C₂×C₃	C₃×D₄²	1	4	yes	—
420	C₂×C₃	C₂×C₃×AGL(1,ℤ/8)	1	4	yes	—
421	C₂×C₃	C₃×#576	1	4	yes	—
422	C₂²	—	2	2	no	—
423	C₂²	—	2	2	no	—
424	C₂²	—	2	2	no	—
425	C₂²	—	2	2	no	—
426	C₂²	—	4	1	no	—
427	C₂²	—	4	1	no	—
428	C₂²	—	4	2	no	—
429	C₂²	—	4	2	no	—
430	C₂²	—	4	2	no	—
431	C₂²	—	4	2	no	—
432	C₂²	—	4	2	no	—
433	C₂²	—	4	2	no	—
434	C₂²	—	8	8	no	—
435	C₂²	C₂²×C₃²×D₄	1	1	yes	characteristic
436	C₄	—	2	1	no	—
437	C₄	—	2	2	no	—
438	C₄	—	2	2	no	—
439	C₄	—	4	1	no	—
440	C₄	—	4	1	no	—
441	C₄	C₃²×AGL(1,ℤ/8)	1	2	yes	complemented
442	C₄	C₂²×C₃²×D₄	1	1	yes	characteristic
443	C₄	C₃²×H(5,𝔽₂)	1	1	yes	characteristic
444	C₃	C₃×#3274	1	4	yes	direct product factor
445	C₂	—	4	2	no	—
446	C₂	—	4	2	no	—
447	C₂	—	8	2	no	—
448	C₂	—	8	2	no	—
449	C₂	C₃²×D₄²	1	1	yes	characteristic
450	C₂	C₂×C₃²×AGL(1,ℤ/8)	1	1	yes	characteristic
451	C₂	C₃²×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄×D₄ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₃²×AGL(1,ℤ/8)	C₂	semidirect
2. C₃²×#550	C₂	semidirect
3. C₃²×D₄²	C₂	semidirect
4. C₃²×#493	C₂	semidirect
5. C₃²×#469	C₂	semidirect
6. C₃²×#460	C₂	semidirect
7. C₃²×#459	C₂	semidirect
8. C₃²×#449	C₂	semidirect
9. C₃²×#447	C₂	semidirect
10. C₃²×C₄×SD₁₆	C₂	semidirect
11. C₃²×#436	C₂	semidirect
12. C₃×#3274	C₃	direct (N × H)
13. C₂×C₃²×#41	C₂²	semidirect
18. C₂×C₃²×SD₁₆	C₂²	semidirect
20. C₂×C₃²×D₈	C₂²	semidirect
21. C₂×C₃²×D₈	C₂²	semidirect
22. C₂×C₃²×M₁₆	C₂²	semidirect
25. C₃²×#127	C₂²	semidirect
29. C₃²×#121	C₂²	semidirect
31. C₃²×C₄×Q₈	C₂²	semidirect
35. C₃²×#106	C₂²	semidirect
36. C₃²×C₄⋊C₈	C₂²	semidirect
37. C₃²×#103	C₂²	semidirect
39. C₃²×#102	C₂²	semidirect
40. C₃²×#102	C₂²	semidirect
41. C₃²×C₂²⋊C₈	C₂²	semidirect
42. C₃²×C₄×C₈	C₂²	semidirect
43. C₂×C₃×AGL(1,ℤ/8)	C₂×C₃	semidirect
44. C₃×#550	C₂×C₃	semidirect
45. C₃×D₄²	C₂×C₃	semidirect
46. C₃×#493	C₂×C₃	semidirect
47. C₃×#469	C₂×C₃	semidirect
48. C₃×#460	C₂×C₃	semidirect
49. C₃×#459	C₂×C₃	semidirect
50. C₃×#449	C₂×C₃	semidirect
51. C₃×#447	C₂×C₃	semidirect
52. C₃×C₄×SD₁₆	C₂×C₃	semidirect
53. C₃×#436	C₂×C₃	semidirect
77. C₃²×SD₁₆	D₄	semidirect
95. #3274	C₃²	direct (N × H)
96. C₂×C₃×#41	C₂²×C₃	semidirect
101. C₂×C₃×SD₁₆	C₂²×C₃	semidirect
103. C₂×C₃×D₈	C₂²×C₃	semidirect
104. C₂×C₃×D₈	C₂²×C₃	semidirect
105. C₂×C₃×M₁₆	C₂²×C₃	semidirect
108. C₃×#127	C₂²×C₃	semidirect
112. C₃×#121	C₂²×C₃	semidirect
114. C₃×C₄×Q₈	C₂²×C₃	semidirect
118. C₃×#106	C₂²×C₃	semidirect
119. C₃×C₄⋊C₈	C₂²×C₃	semidirect
120. C₃×#103	C₂²×C₃	semidirect
122. C₃×#102	C₂²×C₃	semidirect
123. C₃×#102	C₂²×C₃	semidirect
124. C₃×C₂²⋊C₈	C₂²×C₃	semidirect
125. C₃×C₄×C₈	C₂²×C₃	semidirect
134. C₃²×Q₈	C₂×D₄	semidirect
163. C₃²×C₈	C₂×D₄	semidirect
164. C₂×AGL(1,ℤ/8)	C₂×C₃²	semidirect
165. #550	C₂×C₃²	semidirect
166. D₄²	C₂×C₃²	semidirect
167. #493	C₂×C₃²	semidirect
168. #469	C₂×C₃²	semidirect
169. #460	C₂×C₃²	semidirect
170. #459	C₂×C₃²	semidirect
171. #449	C₂×C₃²	semidirect
172. #447	C₂×C₃²	semidirect
173. C₄×SD₁₆	C₂×C₃²	semidirect
174. #436	C₂×C₃²	semidirect
198. C₃×SD₁₆	C₃×D₄	semidirect
235. C₃²×C₄	AGL(1,ℤ/8)	semidirect
238. C₂×#41	C₂²×C₃²	semidirect
243. C₂×SD₁₆	C₂²×C₃²	semidirect
245. C₂×D₈	C₂²×C₃²	semidirect
246. C₂×D₈	C₂²×C₃²	semidirect
247. C₂×M₁₆	C₂²×C₃²	semidirect
250. #127	C₂²×C₃²	semidirect
254. #121	C₂²×C₃²	semidirect
256. C₄×Q₈	C₂²×C₃²	semidirect
260. #106	C₂²×C₃²	semidirect
261. C₄⋊C₈	C₂²×C₃²	semidirect
262. #103	C₂²×C₃²	semidirect
264. #102	C₂²×C₃²	semidirect
265. #102	C₂²×C₃²	semidirect
266. C₂²⋊C₈	C₂²×C₃²	semidirect
267. C₄×C₈	C₂²×C₃²	semidirect
276. C₃×Q₈	C₂×C₃×D₄	semidirect
305. C₃×C₈	C₂×C₃×D₄	semidirect
336. SD₁₆	C₃²×D₄	semidirect
373. C₃×C₄	C₃×AGL(1,ℤ/8)	semidirect
376. C₃²	#3274	direct (N × H)
385. Q₈	C₂×C₃²×D₄	semidirect
414. C₈	C₂×C₃²×D₄	semidirect
441. C₄	C₃²×AGL(1,ℤ/8)	semidirect
444. C₃	C₃×#3274	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃²×AGL(1,ℤ/8)	2
C₃²×#550	2
C₃²×D₄²	2
C₃²×#493	2
C₃²×#469	2
C₃²×#460	2
C₃²×#459	2
C₃²×#449	2
C₃²×#447	2
C₃²×C₄×SD₁₆	2
C₃²×#436	2
C₃×#3274	3

Subgroups — C32×#3274 #11770323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#3274 #11770323